The General Theory of Homogenization

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of François Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly.

For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered.

• H-measures
• compactness
• continuum mechanics
• differential equation
• homogenization
• partial differential equations
• physics

From the reviews:

“The book is divided in 34 short chapters including the introduction and a conclusion. … Each chapter ends with some historical details on the mathematical authors who have been cited in the chapter. The book is surely interesting both from the mathematical point of view … and for the historical background on homogenization theory, including H-convergence and Young measures. … many mathematicians will learn how to use the key tools and to develop the basic but deep tools which can be used within this context.” (Alain Brillard, Zentralblatt MATH, Vol. 1188, 2010)

“In his book Luc Tartar ‘re-animates’ the historical genesis of one of the most important ideas in mathematical analysis of the last century, leading to a general theory of homogenization … . a treasure trove for all those working in the field and those who want to get acquainted with it. … it encourages young researchers to build on the material offered and develop new ideas. … he or she will appreciate the author’s effort to give a ‘full’ and perhaps necessarily personalized picture.”­­­ (Guenter Leugering, Mathematical Reviews, Issue 2011 c)

• Dept. Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA

  Luc Tartar

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